Simple Equation Method Research Articles

New singular wave solutions to the KdV‐Sawada‐Kotera‐Ramani equation by the modified simple equation method

The singular solutions and some special solutions of the KdV‐Sawada‐Kotera‐Ramani equation which have been analyzed in detail with solution functions supplemented by the relevant 3‐dimensional presentations of them. In the paper, very large number of possible concrete solutions are presented and discussed, based on the very various choices of possible values of the integration constants existing in the general solution function forms of the relevant class of problems being investigated. The contents of the main chapters have been classified and represented in a more concise and informative manner by pointing out the most important characteristics of the relevant solution sub‐results related by the actually chosen sub‐system of the general system of coupled ordinary differential equations and this process motivated this very detailed study. Finally, this paper obtains the unusually widespread collection of all these relatively laborsome solutions' set which can promote the application of the equation.

Bright, periodic, compacton and bell-shape soliton solutions of the extended QZK and (3 + 1)-dimensional ZK equations

The (3 + 1)-dimensional Zakharov–Kuznetsov (ZK) and the new extended quantum ZK equations are functional to decipher the dense quantum plasma, ion-acoustic waves, electron thermal energy, ion plasma, quantum acoustic waves, and quantum Langmuir waves. The enhanced modified simple equation (EMSE) method is a substantial approach to determine competent solutions and in this article, we have constructed standard, illustrative, rich structured and further comprehensive soliton solutions via this method. The solutions are ascertained as the integration of exponential, hyperbolic, trigonometric and rational functions and formulate the bright solitons, periodic, compacton, bell-shape, parabolic shape, singular periodic, plane shape and some new type of solitons. It is worth noting that the wave profile varies as the physical and subsidiary parameters change. The standard and advanced soliton solutions may be useful to assist in describing the physical phenomena previously mentioned. To open out the inward structure of the tangible incidents, we have portrayed the three-dimensional, contour plot, and two-dimensional graphs for different parametric values. The attained results demonstrate the EMSE technique for extracting soliton solutions to nonlinear evolution equations is efficient, compatible and reliable in nonlinear science and engineering.

Two component model of microtubules and continuum approximation

In the present work, we study the nonlinear dynamics of microtubules, the basic components of the eukaryotic cytoskeleton. We introduce a two-component model describing tangential oscillations of dimers. A crucial nonlinear differential equation is solved using continuum approximation. We show that the dynamics of microtubules can be explained in terms of kink and antikink solitary waves. We used two mathematical procedures, that is the tangent hyperbolic function method and, more general, the simplest equation method. It is shown that both procedures bring about equal solutions.

Stable Optical Solitons for the Higher-Order Non-Kerr NLSE via the Modified Simple Equation Method

This paper studies the propagation of the short pulse optics model governed by the higher-order nonlinear Schrödinger equation (NLSE) with non-Kerr nonlinearity. Exact one-soliton solutions are derived for a generalized case of the NLSE with the aid of software symbolic computations. The modified Kudryashov simple equation method (MSEM) is employed for this purpose under some parametric constraints. The computational work shows the difference, effectiveness, reliability, and power of the considered scheme. This method can treat several complex higher-order NLSEs that arise in mathematical physics. Graphical illustrations of some obtained solitons are presented.

New solitary wave and other exact solutions of the van der Waals normal form for granular materials

The investigation of exact solitary wave solutions to the nonlinear partial differential equation plays an important role to understand any physical phenomena in diverse applied fields. The current work is related to the most prominent nonlinear model named as the van der Waals normal form that appeared naturally and also industrially for the granular materials. In oceanography, the sea ice, sand and snow are some examples of aforesaid matter among others. We employ two novel integration approaches named as the simplest equation method and the expa function method to explore the above mentioned van der Waals model. As a backlash, many new solitary waves and other exact solutions are retrieved. The obtained results depict that the used approaches are simple and effective to deal with nonlinear models. Also, the numerical simulation of some solutions via two and three dimension graphical configurations are presented for certainty and exactness.

A new three dimensional dissipative Boussinesq equation for Rossby waves and its multiple soliton solutions

A new three dimensional nonlinear dynamic theoretical model is derived from fluid mechanics system. In this paper, From the quasi-geostrophic barotropic potential vorticity equation, we obtain a three dimensional dissipative Boussinesq equation by the reduced perturbation method, i.e.utt+e1uxx+e2(u2)xx+e3utxy+e4uxxxx+e5uxxyy=0. It is emphasized that the new equation is different from the existing Boussinesq equations, which describe the three dimensional nonlinear Rossby waves in the atmosphere. Moreover, we explore the dispersion relation of the linear wave through the new equation. Using the travelling wave method and simplest equation method, the general solution and soliton solutions of the equation are obtained successfully respectively. Finally, the formation mechanism of Rossby waves is discussed by multiple soliton solutions.

On some novel optical wave solutions to the paraxial M-fractional nonlinear Schrödinger dynamical equation

The focus of this article is to find some new exact solutions to the M-fractional paraxial nonlinear Schrodinger equation with Kerr media by employing the modified simple equation method and the auxiliary equation method. A set of novel travelling wave solutions are observed such as bright, dark, periodic and optical solitons. Moreover, the physical interpretation of nonlinear waves would also be demonstrated with the aid of scientific computing.

New multiple-different impressive perceptions for the solitary solution to the magneto-optic waveguides with anti-cubic nonlinearity

In this study, we will establish multiple-impressive different perceptions for the solitary wave solution to the magneto-optic waveguides with anti-cubic nonlinearity (MOWWACN) in the framework of three distinct manners. The three manners are implemented in the same vein and parallel to extract multiple new perceptions for the solitary wave solutions to this model. The first one is the extended simple equation method (ESEM), while the second one is the Paul-Painlevé approach method (PPAM) and the third one is the Riccati-Bernolli Sub ODE method (RBSODM). A good comparison has been constructed not only between our extracted solitary wave solutions which are achieved by these three methods but also with that extracted previously by other authors.

Solitons and Other Solutions for the Nonlinear Convection–Diffusion–Reaction Equation with Power-Law Nonlinearity by the Extended Simplest Equation Method

Solitons and Other Solutions for the Nonlinear Convection–Diffusion–Reaction Equation with Power-Law Nonlinearity by the Extended Simplest Equation Method

New perception of the exact solutions of the 3D-fractional Wazwaz-Benjamin-Bona-Mahony (3D-FWBBM) equation

In this article we will utilize new perception of the Benjamin-Bona-Mahony equation that represents developed stretch for the Korteweg-de Varies equation which denotes to the unidirectional propagation of small amplitude long waves on the surface of hydro magnetic and acoustic waves in channel significantly for shallow water. The main idea of this research realizing new exact and hence solitary solutions of this new perception with its spatial and temporal variables and fractional order whose called 3D-fractional Wazwaz-Benjamin-Bona-Mahony model (3D -FWBBM). Two important different techniques are invited for the first time to proposed this new perception of the exact solutions, the first one is the modified simple equation method (MSEM) and the second is the Riccati-Bernolli Sub-ODE method (RBSOM) whose are used effectively to obtain accurate description of the wave solutions to this equation.

New vision for the soliton solutions to the complex Hirota-dynamical model

In this paper, the nonlinear complex Hirota-dynamical model NLCHM in which the third derivative term represents the self-interaction in the high-frequency subsystem is established. This model plays a vital role in plasma physics because there are agreements between the self-interaction in the high-frequency and the well- known self-focusing effect in plasma. Many soliton solutions to this equation model have been achieved perfectly using the solitary wave ansatz method (SWAM). Furthermore, in the same vein and related subject the extended simple equation method (ESEM) has been applied perfectly to achieve new perception of soliton solutions to this model. A good isomorphic between the achieved results and that achieved previous by other authors have been listed.

Closed-form solutions and conservation laws of a generalized Hirota–Satsuma coupled KdV system of fluid mechanics

Abstract In this article, a generalized Hirota–Satsuma coupled Korteweg–de Vries (KdV) system is investigated from the group standpoint. This system represents an interplay of long waves with distinct dispersion correlations. Using Lie’s theory several symmetry reductions are performed and the system is reduced to systems of non-linear ordinary differential equations (NLODEs). Subsequently, the simplest equation method is invoked to find exact solutions of the NLODE systems, which then provides the solitary wave solutions for the system under discussion. Finally, we construct conservation laws of generalized Hirota–Satsuma coupled KdV system with the aid of general multiplier approach.

Exact solutions of conformable fractional differential equations

This article is about to formulate exact solutions of the time fractional Dodd-Bullough-Mikhailov (DBM) equation, Sinh-Gordon equation and Liouville equation by utilizing simplest equation method (SEM) in conformable fractional derivative (CFD) sense. Using a proper transformation, the original equations are transformed to nonlinear ordinary differential equations (ODEs). The method is very simple in comparison with the classical techniques and very much effective for solving fractional order partial differential equations (FOPDEs).

Optical soliton perturbation with dual forms of simple equation approach: A transparent comparison

This paper studies dual forms of integrability to reveal a spectrum of optical soliton solutions of a governing model. These are modified simple equation method and extended simple equation scheme. The model also incorporates perturbation terms that are of Hamiltonian type. Four forms of nonlinear refractive index structures are studied.

Symmetry analysis with similarity reduction, new exact solitary wave solutions and conservation laws of (3 + 1)-dimensional extended quantum Zakharov–Kuznetsov equation in quantum physics

A recently defined (3+1)-dimensional extended quantum Zakharov–Kuznetsov (QZK) equation is examined here by using the Lie symmetry approach. The Lie symmetry analysis has been used to obtain the varieties in invariant solutions of the extended Zakharov–Kuznetsov equation. Due to existence of arbitrary functions and constants, these solutions provide a rich physical structure. In this paper, the Lie point symmetries, geometric vector field, commutative table, symmetry groups of Lie algebra have been derived by using the Lie symmetry approach. The simplest equation method has been presented for obtaining the exact solution of some reduced transform equations. Finally, by invoking the new conservation theorem developed by Nail H. Ibragimov, the conservation laws of QZK equation have been derived.

Optical singular and dark solitons to the (2 + 1)-dimensional time–space fractional nonlinear Schrödinger equation

In this study, the (2+1)-dimensional nonlinear Schrödinger equation with fractional-order derivative in both time and space is considered within the modified Riemann–Liouville derivatives. This fractional-order equation is newly derived by using the fractional variational principle and the semi-inverse method. By introducing the fractional complex transform, the (G′∕G)-expansion method, the tanh–coth method, and the modified simple equation method are used to obtain the explicit analytical solutions of this equation. As a result, optical singular and dark soliton solutions are obtained. The behaviours of the soliton solutions are expressed by 2D and 3D graphs.

New explicit and exact traveling wave solutions of (3+1)-dimensional KP equation

<p style='text-indent:20px;'>In this paper, we investigate explicit exact traveling wave solutions of the generalized (3+1)-dimensional KP equation <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation} \ (u_{t}+\alpha uu_{x}+\beta u_{xxx})_{x}+\gamma u_{yy}+\delta u_{zz} = 0, \ \ \ \ \beta&gt;0 \;\;\;\;\;\;(1) \ \end{equation}$ \end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>describing the dynamics of solitons and nonlinear waves in the field of plasma physics and fluid dynamics, where <inline-formula><tex-math id="M1">\begin{document}$ \alpha, \beta, \gamma, \delta $\end{document}</tex-math></inline-formula> are nonzero constants. By using the simplified homogeneous balance method, we get one single soliton solution and one double soliton solution of (1). Moreover, we use the extended tanh method with a Riccati equation and the simplest equation method with Bernoulli equation to obtain seven sets of explicit exact traveling wave solutions. When <inline-formula><tex-math id="M2">\begin{document}$ \delta = 0 $\end{document}</tex-math></inline-formula> or <inline-formula><tex-math id="M3">\begin{document}$ \gamma = 0 $\end{document}</tex-math></inline-formula>, (1) reduces to (2+1)-dimensional KP equation. Therefore, we can get some exact traveling wave solutions of (2+1)-dimensional KP equation.

Symmetry reductions and exact solutions of a two-wave mode Korteweg-de Vrie equation

A two-wave mode Korteweg-de Vries equation is investigated by using Lie symmetry analysis. The similarity reductions and new exact solutions are obtained via the simplest equation method. Exact solutions including solitons are shown. In addition, the conservation laws are derived using the multiplier approach.

Analytical and numerical treatment to the (2+1)-dimensional Date-Jimbo-Kashiwara-Miwa equation

Abstract In this work, we study the (2 + 1)-dimensional Date-Jimbo-Kashiwara-Miwa (DJKM) equation. We employ the extended tanh function method and the simple equation method to achieve analytical soliton solutions. Moreover, numerical treatment for this equation is introduced by the finite difference method. We justify the accuracy of the obtained results by exhibiting illustrative tables and proper graphs.

Three distinct and impressive visions for the soliton solutions to the higher-order nonlinear Schrodinger equation

In this article we will use three different schemas to establish three different visions of the soliton solutions to the higher-order nonlinear Schrodinger equation (HONSE). The first schema is the extended simple equation method (ESEM), while the second one is the Paul-Painlevé approach method (PPAM) and the third one is the Ricatti-Bernolli Sub ODE method (RBSOM). The three methods are used in the same vein and parallel to construct many new soliton solutions to this model. A comparison between the obtained results and that achieved previously has been demonstrated.